Saturday, January 24, 2009

My recent conjecture that the non-differentiable function
$$f(x) = \sum_{i=0}^\infty \frac{sin(2^i x)}{2^i}$$
reaches the maximum at the point \(x_0=36\pi/127\) appears to be false (thanks to people from sci.math group for pointing it out).
This is demonstrated by the following graph of the differential quotient \(\frac{f(x_0)-f(x_0-h)}h\), where \(x_0=\frac{36\pi}{127}\).

As it can be seen from the graphs, at the certain point h between \(10^{-29}\) and \(10^{-28}\) the quotient becomes negative, which means that there exists such h, that \(f(x_0+h) > f(x_0)\). Apparently, the extremal value is h≈1.998e-29, giving \(f(x_0+h) - f(x_0) \approx 3.06\dot10^{-32}\).

f(x0)=

1.329833276287310850440418286206506387707650784...

f(x0-1.998e-29)=

1.329833276287310850440418286206535446963412182...

An important note about this calculation: the usual IEEE double accuracy (53 bits, about 16 decimal digits) is insufficient, and long doubles must to be used to obtain the result. For the above calculation I used 1000-bit long floating point values having roughly roughly 300 decimal digits. This, I believe, makes the numeric results trustworthy enough.
There is still a possibility that the maximum is reached at some rational multiple of π, but it never be as simple, as 36π/127. I feel a strange mix of frustration and enlightenment regarding this failure, the result seems absolutely counter-intuitive for me. Why \(x_0/\pi\) is so incredibly close to the rational value? Is it just a fantastic coincidence, or a consequence of a some deeper facts?
This result strongly resembles the "Almost Integer" values in mathematics. it can be re-formulated in the following way:
$$ N = \frac{127}\pi \mathrm{argmin} \sum_{i=0}^\infty \frac{sin(2^i x)}{2^i} = 36+\epsilon $$
where \( \left| \epsilon \right| < 10^{-27} \). I.e. N is an almost integer value.